On gauge invariant observables for identity-based marginal solutions in bosonic and super string field theory

Transcription

1 On gauge invariant observables for identity-based marginal solutions in bosonic and super string field theory Isao Kishimoto Niigata University, Japan July 31, 2014 String Field Theory and Related Aspects VI, SFT2014 SISSA, Trieste (Italy) 1 / 33

2 References References This talk is based on I. K. and Tomohiko Takahashi, Comments on observables for identity-based marginal solutions in Berkovits superstring field theory, JHEP07(2014)031 [arxiv: ] and references therein: I. K. and T. T. (2005, 2013); Shoko Inatomi, I. K. and T. T. (2012). cf. Erler, Ishibashi and Maccaferri s talks 2 / 33

3 Contents Contents 1 Introduction and summary 2 Tachyon vacuum around the identity-based marginal solution Identity-based marginal solutions Deformed algebra Tachyon vacuum solution Energy and gauge invariant overlaps 3 Evaluation of observables for identity-based marginal solutions Interpolation On gauge equivalence relations Energy and gauge invariant overlaps 4 Conclusions 3 / 33

4 Introduction and summary Introduction and summary In my previous we have constructed [IKT(2012)] nontrivial solutions around identity-based marginal solutions in modified cubic SSFT using the G K Bcγ algebra. Some time ago in [KT(2005)], we constructed identity-based marginal solution Φ J in Berkovits WZW-like SSFT. Recntly, Erler has constructed the tachyon vacuum solution Φ E T in Berkovits WZW-like SSFT. We will construct the tachyon vacuum solution Φ T around the identity-based marignal solution Φ J, using a version of the extended KBc algebra in the framework of Berkovits WZW-like SSFT. 1 / 33

5 Introduction and summary It is difficult to evaluate vacuum energy and gauge invariant overlaps for identity-based solutions because of singular property of the identity state: I ( ) I, corresponding to zero width in the sliver frame, is indefinite at least naively. Instead of straightforward calculations, there have been various studies about the gauge invariants indirectly for identity-based solutions in bosonic SFT. (2001, cf. Takahashi s talks) On the other hand, gauge invariants for wedge-based solutions using KBc algebra and its extention have been evaluated exactly. ([Schnabl(2005)] ) 2 / 33

6 Introduction and summary In our previous paper [KT(2013)], we have evaluated the gauge invariant overlaps (GIO), Ψ V, for identity-based marginal solution Ψ J in bosonic SFT: Ψ J V = Ψ ES T V Ψ T V using the relationship to (wedge-based) tachyon vacuum solutions: 1 Ψ J = Ψ ES T Ψ T + Q Ψ t T Λ t dt. 0 In fact, we can show that Ψ ES T [Erler-Schnabl(2009)] and the sum of Ψ J [Takahashi-Tanimoto(2001)] and Ψ T [IKT(2012)] are gauge equivalent: ( 1 ) Ψ J + Ψ T = g 1 Ψ ES T g + g 1 Q B g, g = P exp Λ t dt 0 S[Ψ J ; Q B ] = S[Ψ ES T ; Q B ] S[Ψ T ; Q ΨJ ] = 0 3 / 33

7 Introduction and summary Figure : Identity/wedge-based solutions in bosonic SFT 4 / 33

8 Introduction and summary extention to SSFT 1 e Φ J Φ ˆQe J = e ΦE T Φ ˆQe E T e Φ T ˆQΦJ e Φ T + ˆQ ΦT (t) Λ t dt 0 Using this relation among identity-based marginal solution Φ J and wedge-based tachyon vacuum solutions, Φ E T and Φ T, the GIO for identity-based marginal solution can be evaluated: Φ J V = Φ E T V Φ T V. Actually, Φ E T and log(eφ J e Φ T ) are gauge equivalent and we have S[Φ J ; ˆQ] = S[Φ E T ; ˆQ] S[Φ T ; ˆQ ΦJ ] = 1/(2π 2 ) 1/(2π 2 ) = 0. The energy for the identity-based marginal solution Φ J is zero, which agrees with the previous result as a consequence of ξ zeromode counting [KT(2005)]. 5 / 33

9 Introduction and summary Figure : Identity/wedge-based solutions in Berkovits WZW-like SSFT including GSO( ) sector 6 / 33

10 Tachyon vacuum around the identity-based marginal solution Contents 1 Introduction and summary 2 Tachyon vacuum around the identity-based marginal solution Identity-based marginal solutions Deformed algebra Tachyon vacuum solution Energy and gauge invariant overlaps 3 Evaluation of observables for identity-based marginal solutions Interpolation On gauge equivalence relations Energy and gauge invariant overlaps 4 Conclusions 7 / 33

11 Tachyon vacuum around the identity-based marginal solution Identity-based marginal solutions Identity-based marginal solutions Identity-based marginal solution in Berkovits WZW-like SSFT: Φ J = Ṽ L a (F a )I, ṼL a dz (f) 2πi f(z) 1 cγ 1 ψ a (z), 2 C L γ 1 (z) = e ϕ ξ(z). F a ( 1/z) = z 2 F a (z), C L : a half unit circle: z = 1, Re z 0 I: the identity state, ψ a : matter worldsheet fermion. ψ a (y)ψ b (z) 1 y z J a (y)j b (z) 1 2 Ωab, J a (y)ψ b (z) 1 y z f ab c ψc (z), 1 1 (y z) 2 2 Ωab + 1 y z f ab cj c (z), Ω ab = Ω ba, f ab c Ωcd + f ad c Ωcb = 0, f ab c = f ba c, f ab d f cd e + f bc d f ad e + f ca d f bd e = 0. EOM in the NS sector is satisfied: η 0 (e Φ J Q B e Φ J ) = 0. 8 / 33

12 Tachyon vacuum around the identity-based marginal solution Identity-based marginal solutions By expanding the NS action S[Φ; Q B ] of Berkovits WZW-like SSFT around Φ J as e Φ = e Φ J e Φ, we have S[Φ; Q B ] = S[Φ J ; Q B ] + S[Φ ; Q ΦJ ], where S[Φ ; Q ΦJ ] is given by a deformed BRST operator: Q ΦJ = Q B V a (F a ) Ωab C(F a F b ). V a (F a ) and C(F a F b ) are given by integrations along the whole unit circle: dz V a 1 dz (f) f(z)(cj a (z) + γψ a (z)), C(f) 2πi 2 2πi f(z)c(z). 9 / 33

13 Tachyon vacuum around the identity-based marginal solution Deformed algebra Deformed algebra A version of the extended KBc algebra with Q B Q Q ΦJ L n L n = {Q, b n } = L n 1 2 k Z F a,kj a n k Ωab k Z F a,n kf b,k F a,n dσ 2π ei(n+1)σ F a(e iσ ), Relations among string fields K, B, c, γ: B 2 = 0, c 2 = 0, Bc + cb = 1, BK = K B, F a,n = ( 1) n F a, n K c ck = Kc ck c, γb + Bγ = 0, cγ + γc = 0, K γ γk = Kγ γk γ, ˆQ B = K, ˆQ K = 0, ˆQ c = ck c γ 2 = ckc γ 2 = c c γ 2, ˆQ γ = ˆQγ = c γ 1 2 ( c)γ where ˆQ Q σ 3, ˆQ Q B σ 3 and σ i are Pauli matrices (CP factor) for the GSO( ) sector B = π 2 BL 1 Iσ 3, c = 2 π Û1 c(0) 0 σ 3, γ = 2 π Û1 γ(0) 0 σ 2 K = π 2 K L 1 I, K = π 2 KL 1 I 10 / 33

14 Tachyon vacuum around the identity-based marginal solution Deformed algebra For the string fields γ 1, ζ, V, we have γ 1 γ = γγ 1 = 1, γ 1 B + Bγ 1 = 0, γ 1 c + cγ 1 = 0, K γ 1 γ 1 K = Kγ 1 γ 1 K γ 1, ˆQ γ 1 = ˆQγ 1 = c γ ( c)γ 1, ˆQ ζ = ˆQζ = cv + γ where γ 1 = π 2 Û1 γ 1 (0) 0 σ 2, ζ = γ 1 c = V = 1 2 γ 1 c = π 2 Û1 1 2 γ 1 c(0) 0 iσ 1. 2 π Û1 γ 1 c(0) 0 iσ 1, K, B, c, γ, γ 1, ζ, V and ˆQ have the same algebraic structure as that of the extended KBc algebra with ˆQ [Erler(2013)]. 11 / 33

15 Tachyon vacuum around the identity-based marginal solution Tachyon vacuum solution From the result in [Erler(2013)] and the above algebra, we can immediately construct a solution Φ T in the theory with ˆQ : ( ) e Φ T B = 1 c 1 + K + q ζ + ( ˆQ B ζ) 1 + K (q is a nonzero constant.) Actually, Φ T satisfies e Φ T ˆQ e Φ T = c ( ˆQ B c) = (c K ˆQ 1 (Bc)) 1 + K which is in the small Hilbert space, and therefore the EOM in the NS sector holds: ˆη(e Φ T ˆQ e Φ T ) = 0 (ˆη η 0 σ 3 ) 12 / 33

16 Tachyon vacuum around the identity-based marginal solution Tachyon vacuum solution Expanding the action S[Φ ; ˆQ ] around the solution Φ T as e Φ = e Φ T e Φ, we have a new BRST operator ˆQ Φ T : ˆQ Φ T Ξ = ˆQ Ξ + (e Φ T ˆQ e Φ T )Ξ ( 1) Ξ Ξ(e Φ T ˆQ e Φ T ). Note that e Φ T ˆQ e Φ T is the tachyon vacuum solution on the marginally deformed background in the modified cubic SSFT. We can find a homotopy operator  for ˆQ Φ T : Â Ξ = 1 2 (A Ξ + ( 1) Ξ Ξ A ), such as { ˆQ Φ T,  } = 1, ( ) 2 = 0, where A B 1 + K is a homotopy state in the small Hilbert space. No physical open string state around the solution Φ T the tachyon vacuum solution in the marginally deformed background. 13 / 33

17 Tachyon vacuum around the identity-based marginal solution Tachyon vacuum solution Figure : Identity/wedge-based solutions and BRST operators around them. ˆQ ΦT has no cohomology in the small Hilbert space. 14 / 33

18 Tachyon vacuum around the identity-based marginal solution Energy and gauge invariant overlaps Energy and gauge invariant overlaps NS action S[Φ ; ˆQ ΦJ ] around Φ J : 1 S[Φ ; ˆQ [ ΦJ ] = dt Tr (ˆη(g(t) 1 t g(t)) ) ] (g(t) 1 ˆQΦJ g(t)). 0 g(t): an interpolating string field s.t. g(0) = 1 and g(1) = e Φ. Tr A 1 2tr I A ; tr : trace for the Chan-Paton factor for GSO(±) sector We take an interpolating string field as g T (t) = 1 + t(e Φ T 1) and the integrand in the action for the solution: S[Φ T ; ˆQ ΦJ ], can be manipulated in the same way as the Erler solution, with Q B Q Q ΦJ, L n L n. As a result, we have [ Tr (ˆη(gT (t) 1 t g T (t)) ) ] (g T (t) 1 ˆQΦJ g T (t)) where = 2q2 t 2 (1 t)(2q 2 t 1) 1 (1 t + q 2 t 2 ) 3 dθ(1 θ)x (θ) + q2 t(1 t) 1 0 (1 t + q 2 t 2 ) 2 dθx (θ) 0 [ X (θ) = Tr B(ˆη(cV ))e θk cv e (1 θ)k ]. 15 / 33

19 Tachyon vacuum around the identity-based marginal solution Energy and gauge invariant overlaps We use the result for the modified cubic SSFT in the marginally deformed background [IKT(2012)]: ( e αk = e α π π α 2 C Û α+1 T exp du dvf a (v) 4 J a (iv + π ) 0, α 4 u) f a (v) F a(tan(iv + π 4 )) 2π 2 cos 2 (it + π 4 ), C π dv Ω ab f a (v)f b (v), 2 where T is an ordering symbol with respect to the real part of the argument of J a. Finally, the trace can be evaluated as Tr [B(ˆη(cV ))e θk cv e (1 θ)k ] = 1 (η 0 γ 1 ( π 4 2 ))γ 1 ( π B 2 (1 2θ)) 1 L c c( π ξηϕ 2 )c c(π 2 (1 2θ)) ( π ) e π 2 exp C 2 du dvf a (v)j a (2iv + u) π 2 mat bc 16 / 33

20 Tachyon vacuum around the identity-based marginal solution Energy and gauge invariant overlaps The last factor in the trace, which comes from the matter sector, is 1: ( π ) e π 2 exp C 2 du dvf a (v)j a (2iv + u) = 1 π 2 as proved in [IKT(2012)], and so the trace becomes the same result as the case of the Erler solution. Consequently, the vacuum energy is unchanged from the case in the original background without marginal deformations, namely, E = S[Φ T ; ˆQ ΦJ ] = 1/(2π 2 ). mat 17 / 33

21 Tachyon vacuum around the identity-based marginal solution Energy and gauge invariant overlaps Next, we will evaluate the gauge invariant overlap (GIO) in Berkovits WZW-like SSFT. We define the GIO Φ V as Φ V Tr[V(i)Φ]. V(i) : a midpoint insertion of a primary closed string vertex operator with picture #: 1, ghost #: 2, conformal dim.: (0, 0), BRST invariant in the small Hilbert space: [Q B, V(i)] = 0, [η 0, V(i)] = 0. Then, Λ, Ξ ˆQΛ V = 0, ˆηΛ V = 0, ΛΞ V = ( ) Λ Ξ ΞΛ V. Infinitesimal gauge transformation: δ Λ e Φ = ( ˆQΛ 0 )e Φ + e ΦˆηΛ 1 (Λ 0, Λ 1 : gauge parameter string field with the picture # 0, 1.) Namely, δ Λ Φ= ad Φ e ad ˆQΛ0 + ad Φ Φ 1 e ad ˆηΛ 1 = Φ 1 (ad B (A) [B, A] = BA AB) n=0 B n n! (ad Φ) n ( ˆQΛ 0 +( 1) nˆηλ 1 ) 18 / 33

22 Tachyon vacuum around the identity-based marginal solution Energy and gauge invariant overlaps Thanks to the above properties, the GIO is invariant under this gauge transformation: δ Λ Φ V = 0. Inserting 1 = {Q B, ξy (i)}, (Y (z) = c ξe 2ϕ (z) the inverse picture changing operator) the GIO can be rewritten as Φ V = Tr[V(i){Q B, ξy (i)}φ] = Tr[ξY V(i)σ 3 ˆQΦ] = Tr[ξY V(i)σ3 ˆQ Φ] = Tr[ξY V(i)σ 3 e Φ ˆQ e Φ ]. [ ] 1 GIO for the tachyon vacuum Φ T : Φ T V = Tr ξy V(i)σ 3 c 1 + K In a similar way to the calculation of the vacuum energy, we obtain an expression of the GIO in the marginally deformed background: ( Φ T V = e πc ξy V(i )c( π π ) π 2 ) exp 2 du J (u) J (u) = π 2 dv f a (v)j a (iv + u) C π 19 / 33

23 Evaluation of observables for identity-based marginal solutions Contents 1 Introduction and summary 2 Tachyon vacuum around the identity-based marginal solution Identity-based marginal solutions Deformed algebra Tachyon vacuum solution Energy and gauge invariant overlaps 3 Evaluation of observables for identity-based marginal solutions Interpolation On gauge equivalence relations Energy and gauge invariant overlaps 4 Conclusions 20 / 33

24 Evaluation of observables for identity-based marginal solutions Interpolation Interpolation Let us calculate two observables, the vacuum energy and the GIO, for the identity-based marginal solutions. An interpolation: Φ J (t) = tφ J s.t. Φ J (0) = 0, Φ J (1) = Φ J. Φ J (t) : a replacement of weighting function: F a (z) tf a (z) in Φ J. EOM is satisfied: ˆη(e Φ J (t) ˆQe Φ J (t) ) = 0 A new BRST operator Q ΦJ (t) for the theory around Φ J (t): Q ΦJ (t) = Q B tv a (F a ) + t2 8 Ωab C(F a F b ) Following the same procedure as before, we define a string field K (t) ˆQ ΦJ (t)b ( ˆQ ΦJ (t) Q ΦJ (t)σ 3 ). Then, we can construct a tachyon vacuum solution Φ T (t) as ( ) e Φ T (t) B = 1 c 1 + K (t) + q ζ + ( ˆQ B ΦJ (t)ζ) 1 + K (t) 21 / 33

25 Evaluation of observables for identity-based marginal solutions Interpolation It satisfies the EOM around the identity-based solution Φ J (t): ˆη(e Φ T (t) ˆQΦJ (t)e Φ T (t) ) = 0. In particular, Φ T (t) satisfies Φ T (1) = Φ T and Φ T (0) = Φ E T solution (2013)) because Q ΦJ (1) = Q ΦJ and Q ΦJ (0) = Q B. (the Erler Using the above string fields, we define a string field Φ T (t) with the parameter t as e Φ T (t) e Φ J (t) e Φ T (t) and then we find a relation: e Φ T (t) ˆQe ΦT (t) = e Φ J (t) ˆQe Φ J (t) + e Φ T (t) ˆQΦJ (t)e Φ T (t). Hence, Φ T (t) satisfies the EOM of the original theory: ˆη(e Φ T (t) ˆQe ΦT (t) ) = / 33

26 Evaluation of observables for identity-based marginal solutions Interpolation Expanding around the solution Φ T (t) in the theory with ˆQ, we have the theory with the deformed BRST operator ˆQ ΦT (t) : ˆQ ΦT (t) Ξ = ˆQΞ + (e Φ T (t) ˆQe ΦT (t) )Ξ ( 1) Ξ Ξ(e Φ T (t) ˆQe ΦT (t) ) = ˆQ ΦJ (t)ξ + (e Φ T (t) ˆQΦJ (t)e Φ T (t) )Ξ ( 1) Ξ Ξ(e Φ T (t) ˆQΦJ (t)e Φ T (t) ). The last expression implies that ˆQ ΦT (t) is the same as the BRST operator ˆQ Φ T (t) in the theory around Φ T (t), which is a tachyon vacuum solution in the theory around Φ J (t). Following the previous results with appropriate replacement, we find that there exists a homotopy state: A (t) such as ˆQ ΦT (t) A (t) = 1, which implies that B 1+K (t) there is no cohomology for ˆQ ΦT (t) in the small Hilbert space. 23 / 33

27 Evaluation of observables for identity-based marginal solutions Interpolation Figure : Interporating identity/wedge-based solutions and BRST operators around them. With e Φ T (t) e Φ J (t) e Φ T (t), a BRST operator around Φ T (t) ˆQ Φ T (t) = ˆQ ΦT (t) has no cohomology in the small Hilbert space. 24 / 33

28 Evaluation of observables for identity-based marginal solutions Interpolation Differentiating an identity: ˆQ(e Φ T (t) ˆQe ΦT (t) ) + (e Φ T (t) ˆQe ΦT (t) ) 2 = 0 with respect to t, we have ˆQ ΦT (t) d (e Φ T (t) ˆQe ΦT (t) ) = 0. dt Therefore, there exists a state Λ t in the small Hilbert space such as Integrating the above, we have d dt (e Φ T (t) ˆQe ΦT (t) ) = ˆQ ΦT (t) Λ t. e Φ T (1) ˆQe ΦT (1) = e ΦE T ˆQe Φ E T ˆQ ΦT (t) Λ t dt. This relation implies that Φ T (1) = log(e Φ J e Φ T ) is gauge equivalent to the Erler solution Φ E T. 25 / 33

29 Evaluation of observables for identity-based marginal solutions On gauge equivalence relations On gauge equivalence relations Here, we discuss some gauge equivalence relations in terms of the NS sector of Berkovits WZW-like SSFT. A gauge transformation of the superstring field Φ by group elements h(t) and g(t) with one parameter t s.t. g(0) = h(0) = 1 is e Φ(t) = h(t) e Φ g(t), Q B h(t) = η 0 g(t) = 0. (1) For the string fields, Φ and Φ(t), one-form string fields are defined by Ψ e Φ Q B e Φ and Ψ(t) e Φ(t) Q B e Φ(t). From (1), these turn out to be related by a transformation: Ψ(t) = g(t) 1 Q B g(t) + g(t) 1 Ψ g(t), η 0 g(t) = 0. (2) It is the same form as gauge transformations in the modified cubic SSFT. Conversely, given the relation (2), we find that the relation (1) holds for h(t) = e Φ(t) g(t) 1 e Φ. In fact, Q B (e Φ(t) g(t) 1 e Φ ) = 0 holds from (2) 26 / 33

30 Evaluation of observables for identity-based marginal solutions On gauge equivalence relations Differentiating (2) with respect to t and integrating it again, we find another relation between Ψ and Ψ(t): t Ψ(t) = Ψ + Q Φ(t )Λ(t ) dt, η 0 Λ(t) = 0, (3) 0 where Λ(t) = g(t) 1 d dt g(t) and Q ϕ is a modified BRST operator associated with ψ e ϕ Q B e ϕ : Q ϕ λ = Q B λ + ψλ ( 1) λ ψλ. Conversely, supposing that the equations (3) for a given Λ(t) hold, we find the relations (2) hold for the group element g(t) such as g(0) = 1: ( t ) g(t) = P exp Λ(t )dt, 0 where P exp means a t-ordered exponent. Consequently, the above relations (1), (2) and (3) are all equivalent. (We can include internal Chan-Paton factors in these relations.) 27 / 33

31 Evaluation of observables for identity-based marginal solutions Energy and gauge invariant overlaps Energy and gauge invariant overlaps From the gauge equivalence: log(e Φ J e Φ T ) Φ E T we can analytically evaluate gauge invariants for the identity-based marginal solutions. The value of the action: S[Φ J ; ˆQ] + S[Φ T ; ˆQ ΦJ ] = S[Φ E T ; ˆQ] E = S[Φ J ; ˆQ] = S[Φ T ; ˆQ ΦJ ] S[Φ E T ; ˆQ] = 0 The result agrees with the previous one derived from ξ zeromode counting: S[Φ J, Q B ] = 1 0 dt Tr[(η 0 Φ J ) (e tφ J Q B e tφ J )] = 0 28 / 33

32 Evaluation of observables for identity-based marginal solutions Energy and gauge invariant overlaps Evaluation of the GIO for the identity-based marginal solution Φ J V : Using an identity, we have obtained e Φ T (1) ˆQe ΦT (1) = e Φ J ˆQe Φ J + e Φ T ˆQΦJ e Φ T, e Φ J Φ ˆQe J = e ΦE T Φ ˆQe E T e Φ T ˆQΦJ e Φ T + It leads to a relation for the GIOs: 1 0 ˆQ ΦT (t) Λ t dt. Φ J V = Φ E T V Φ T V = 1 ξy V(i )c( π ( {1 π π 2 ) e πc 2 exp π 2 )} du J (u). C π 29 / 33

33 Evaluation of observables for identity-based marginal solutions Energy and gauge invariant overlaps In the same way as the case of bosonic SFT [KT(2013)], it can be rewritten as a difference between two disk amplitudes with the boundary deformation by taking F a (z; s) = 2λ as(1 s 2 ) 1 + z 2 arctan 1 s 2 (z 2 + z 2 ) + s 4 2s 1 s 2 for the function F a (z) in Φ J, which satisfies dz 2πi F a(z; s) = 2λ a π, C L F a (z; s) 4λ a {δ(θ) + δ(π θ)}, (s 1, z = e iθ ). Because the form of weighting function F a except the half-integration mode can be changed by a kind of gauge transformation [KT(2005)], we have the same value of the GIO for a fixed value of λ a, which corresponds to a marginal deformation parameter). 30 / 33

34 Evaluation of observables for identity-based marginal solutions Energy and gauge invariant overlaps Namely, for the limit s 1, the GIO is expressed as Φ J V = 1 ξy V(i )c( π ( {1 π π 2 ) e πc 2 exp π λ 2 a π 2 )} du J a (u). C π In this expression, C becomes divergent for the function lim s 1 F a (z; s) and then it cancels the contact term divergence due to singular OPE among the currents. This expression of the GIO corresponds to the result in [Ellwood(2008)] for a wedge-based marginal solution. 31 / 33

35 Conclusions Conclusions We have applied the method in [IKT(2012)] for cubic (S)SFT to the Erler solution Φ E T for Berkovits WZW-like SSFT. We have constructed a tachyon vacuum solution Φ T around the identity-based marginal solution Φ J [KT(2005)] in SSFT with an extended KBc algebra in the marginally deformed background. Around Φ T, we have obtained a homotopy operator and evaluated vacuum energy and gauge invariant overlap (GIO) for it. The energy is the same value as that on the original background, but the GIO is deformed by the marginal operators. 32 / 33

36 Conclusions Using the above, we have extended our computation for bosonic SFT [KT(2013)] to superstring: We have evaluated the energy and the GIO for the identity-based marginal solution Φ J in the framework of Berkovits WZW-like SSFT. The gauge equivalence relation between Φ E T and log(eφ J e Φ T ) is essential. It is derived from the vanishing cohomology in the small Hilbert space around the interpolating tachyon vacuum solutions. The energy for Φ J vanishes and it is consistent with our previous result using ξ zeromode counting. The GIO for Φ J is expressed by a difference of those of the tachyon vacuum solutions on the undeformed and deformed backgrounds. We hope that this approach to identity-based solutions will be useful to deeply understand bosonic and super SFT. 33 / 33